Doing so would not force users to divulge codes over the phone, and enable support staff to verify identity all without training users that reading codes over the phone is acceptable.
Thoughts on that?
I think if the war against phishing online has taught us anything, it's that humans can't be trusted to not reveal secrets to scammers. Only machine-to-machine public key authentication (like TLS or WebAuthn or U2F) is truly phish-proof.
Note, however, that those are two "totally different sequences of steps" to you and I, and "completely analogous / equivalent sequences of steps" to my father in law :-/
I assume in the case where the customer initiates the call and support is verifying their identity via SMS, they use different text (i.e. not "to confirm you're signing in"). Otherwise, that'd be pretty ridiculous.
the verbiage is the same.
The key situation for giving out an SMS code that the gp is pointing out is the customer initiates the call to the support center.
For example, suppose somebody wants to add a credit-card to their smartphone digital wallet. They have to call the bank issuing their credit-card to do that. Once the customer support person answers the call, a common security verification (e.g. Chase Bank does this) is for them to send you a 6 digit code to your phone. You then repeat this code back to the support person on the call. They want proof of your identity and also proof that you physically have the smartphone with you. Repeating the SMS code to the customer support person is safe because the customer called the official 1-800 number on the back of their card.
That's a totally different sequence of steps from receiving a random call from somebody claiming they are from Chase Bank. Yes, in those cases, you never give out SMS codes to that untrusted person on the phone.